When we talk about Doctor Who there are the original run and "fake" one.
From original run my favorite is 3rd Doctor and his episodes were the best era. It is the early example of the core Doctor Who experience that you get to learn lots of stuff, lore gets expanded, time lords drama, other time lords comes in, the Earth has the usual situations via the UNIT and all. In other Doctors the show gets more of "same shit, but more childish".
From the "fake" series I would say 10th Doctor because the guy who play as the Doctor always wanted to be the Doctor therefore he really plays good (after all best Doctor would be the die hard fan lol), but new generation BS drags down this new series. At least episodes were decent until Steven Moffat killed entire BBC with fake Sherlock Holmes too lol. It was already a fake Doctor Who series and it got more fake the moment 11th Doctor was shit into your TV lol.
My favorite episode: The Three Doctors (1972). As the name indicates watching 3 Doctors in one episode is too much for our human brains lolol. Later on they tried to pull off the same shit but it wasn't as magical as the The Three Doctors episode.
However gotta say Doctor Who was okay but after the 3rd Doctor it kinda got boring which only later on they started to finally get into "Doctor Who" mood. They started to get more parody of our current world by making fun of it and it got more political and all. They started to get the sense of "adventure" further into the fun route. The 3rd Doctor has great episodes that sadly that "style" was killed in favor of kids so 4th Doctor meant the dead of the old Doctor Who lol. So what to like about the "fake" Doctor Who series until the 11th Doctor is the way they took 3rd Doctor style and made it more fun, later on Steven Moffat turned the series into Doctor Who for girls BS and currently it suffers from the same shit Doctor Who suffered from until 7th Doctor.
My favourite villain: I hate the old-fashioned idea of a villain therefore I will skip to John Simm's The Master as the best one lol:
Why?: In the original series The Master was just a simple "bad guy" that every time he showed up you would be like "damn this fucking bug again" lol but surprisingly in the "fake" series they turned him into a victim so not a bad guy but just... The Master lol. Of course the Master would do Mastering stuff!! lol